Abstract
We demonstrate that the structure of complex second-order strongly elliptic operators H on Rd with coefficients invariant under translation by Zd can be analyzed through decomposition in terms of versions Hz, z ∈ Td, of H with z-periodic boundary conditions acting on L2(Id) where I = [0,1〉. If the semigroup S generated by H has a Hõlder continuous integral kernel satisfying Gaussian bounds then the semigroups Sz generated by the Hz have kernels with similar properties and z → Sz extends to a function on Cd\{0} which is analytic with respect to the trace norm. The sequence of semigroups S(m),z obtained by rescaling the coefficients of Hz by c(x) → c(mx) converges in trace norm to the semigroup Ŝz generated by the homogenization Ĥz of Hz. These convergence properties allow asymptotic analysis of the spectrum of H.
Original language | English |
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Pages (from-to) | 621-650 |
Number of pages | 30 |
Journal | Mathematische Zeitschrift |
Volume | 232 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1999 |